Chapter 6.2, Problem 110E

In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.

110. Show that if $f(x)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n{x}^n}$ is a sum of even powers, that is, a_n= 0 if n is odd, then $F={\displaystyle {\int }_0^{x}f(t)dt}$is a sum of odd powers, while if f is a sum of odd powers, then F is a sum of even powers.